SECTION 7-6 Complex Numbers in Rectangular and Polar Forms
Contents
1. lt 0 x 180 for Problems 11 and 12 choose 0 in radians lt 0 5 A 2e B V 20 C Agri lt m Compute the modulus and arguments for parts A and B exactly compute the modulus and argument for part C to two decimal places 9 A V3 i B l i C 5 6i 10 A 1 iV3 B 3i 7 di l A iv3 B V3 i C 8 5i 12 A V3 i B 2 2i CO 6 5i In Problems 13 16 change parts A C to rectangular form Compute the exact values for parts A and B for part C compute a and b for a bi to two decimal places 13 A 2e B V2e i C 3 08e244 14 A 26 B V2eC 9 C 5 71 e748 15 A 62 B V 7e C 4 09e 12288 B V2e6 In Problems 17 22 find zz and zyz 17 z 7e 5 2 2g 18 z 6e z 3e 19 g 5 55 267 20 z 3e87 z 2e 21 z 3 05e 79 z 11 9427 22 z 7 11e9 9 7 2 666 16 A V3eC v C 6 83e 1055 Simplify Problems 23 268 directly and by using polar forms Write answers in both rectangular and polar forms 6 in degrees 23 1 i 2 2i 24 V3 if 25 1 i 26 V3 iV3 0 iV3 144 1 iv3 EET 5 Vi 29 The conjugate of a biis a bi What is the conjugate of re Explain section f 7 7 De Moivre s Theorem 563 30 How is the product of a complex number z with its conju gate related to the modulus of z Explain 31 Show that r e9 is a cube root of re 32 Show th2. Solution Locate in a complex plane first then if x and y are associated with special angles r and 0 can often be determined by inspection A A sketch shows that z is associated with a special 45 triangle Fig 4 Thus by inspection r V2 0 m 4 not 77 4 and a V2 eoo 2 i sin 2 V 2 eC WADI FIGURE 4 B A sketch shows that z is associated with a special 30 60 triangle Fig 5 Thus by inspection r 2 0 57 6 and 5 ST T 5r 2 cos i sin 6 6 JoSr o i FIGURE 5 C A sketch shows that z is not associated with a special triangle Fig 6 So we proceed as follows r V 5 2 539 To two decimal places 0 vr tan e 2 76 To two decimal places 7 6 Complex Numbers in Rectangular and Polar Forms 559 Thus zz 5 39 cos 2 76 i sin 2 76 5 39eC 2 70 FIGURE 6 Figure 7 shows the same conversion done by a graphing utility with a built in conversion routine FIGURE 7 5 2i 5 39e Matched Problem 2 Write parts A C in polar form 0 in radians 7 lt 0 lt m Compute the modulus and arguments for parts A and B exactly compute the modulus and argument for part C to two decimal places A 1 i B 1 4 iV3 C 3 5i EXAMPLE 3 From Polar to Rectangular Form Write parts A C in rectangular form Compute the exact values for parts A and B for part C compute a and b for a bi to two de
3. 7 6 Complex Numbers in Rectangular and Polar Forms 555 20 knot wind 8 1 e cos 0 for the following values of e and identify each curve as a hyperbola ellipse or a parabola A e 0 6 B e 1 C e 2 m 69 Astronomy A The planet Mercury travels around the sun in an elliptical orbit given approximately by 3 442 X 10 fi 0 206 cos 0 where r is measured in miles and the sun is at the pole Graph the orbit Use TRACE to find the distance from Mercury to the sun at aphelion greatest distance from the sun and at perihelion shortest distance from the sun 65 Sailboat Racing Referring to the figure estimate to the B Johannes Kepler 1571 1630 showed that a line join nearest knot the speed of the sailboat sailing at the follow ing a planet to the sun sweeps out equal areas in space in ing angles to the wind 30 75 135 and 180 equal intervals in time see figure Use this information to determine whether a planet travels faster or slower at aphelion than at perihelion Explain your answer 66 Sailboat Racing Referring to the figure estimate to the nearest knot the speed of the sailboat sailing at the follow ing angles to the wind 45 90 120 and 150 Conic Sections Using a graphing utility graph the equation 8 pan 1 e cos 8 for the following values of e called the eccentricity of the conic and identify each curve as a hyperb
4. a bi is said to be in rectangular form Plotting in the Complex Plane Plot the following complex numbers in a complex plane A 2 31 B 3 51 C 4 D 3i Plot the following complex numbers in a complex plane A 4 t 2i B 2 3i C 5 D 4i On a real number line there is a one to one correspondence between the set of real numbers and the set of points on the line Each real number is associated with exactly one point on the line and each point on the line is associated with exactly one real number Does such a correspondence exist between the set of complex numbers and the set of points in an extended plane Explain how a one to one cor respondence can be established FIGURE 2 Rectangular polar relationship ClTiskPFalar 7 6 Complex Numbers in Rectangular and Polar Forms 557 e Polar Form Complex numbers also can be written in polar form Using the polar rectangular relationships from Section 7 5 Z x ly r cos 0 i sin 0 x rcos 0 and y rsin0 we can write the complex number z x iy in polar form as follows z x iy r cos 8 ir sin 0 r cos 0 i sin 0 1 This rectangular polar relationship is illustrated in Figure 2 In a more advanced treat ment of the subject the following famous equation is established e cos 0 i sin 0 2 where e obeys all the basic laws of exponents Thus equation 1 takes on the form z x yi r cos 0 i sin 0 re 3 We will freely use
5. at r e9 is a square root of re 33 Ifz re show that z r e and z r e What do you think z will be for n a natural number 34 Prove i0 S MEME CUT Ze nee p APPLICATIONS amp 35 Forces and Complex Numbers An object is located at the pole and two forces u and v act on the object Let the forces be vectors going from the pole to the complex numbers 20e and 10e9 respectively Force u has a magnitude of 20 pounds in a direction of 0 Force v has a magnitude of 10 pounds in a direction of 60 A Convert the polar forms of these complex numbers to rectangular form and add B Convert the sum from part A back to polar form C The vector going from the pole to the complex number in part B is the resultant of the two original forces What is its magnitude and direction 36 Forces and Complex Numbers Repeat Problem 35 with forces u and v associated with the complex numbers 8e and 6e respectively De Moivre s Theorem e De Moivre s Theorem n a Natural Number e nth Roots of z Abraham De Moivre 1667 1754 of French birth spent most of his life in London doing private tutoring writing and publishing mathematics He belonged to many prestigious professional societies in England Germany and France and he was a close friend of Isaac Newton Using the polar form for a complex number De Moivre established a theorem that still bears his name for raising complex numbers to natural nu
6. cimal places A z 2e6 9 B z 3e 9 C z 7 19e Solution A x iy 2e979 5 Sr me Sr 2 cos i sin 6 6 A a5 2 2 V3 i B x iy 3e 9 3 cos 60 i sin 60 560 7 Additional Topics in Trigonometry r l9 amp e G 2 13i2FE ect 3 01 6 89 FIGURE 8 7 19e 3 81 6 093 EXPLORE DISCUSS 2 Matched Problem 3 EXPLORE DISCUSS 3 e Multiplication and Division in Polar Form C x iy 7 19e 7 19 cos 2 13 i sin 2 13 3 81 6 091 Figure 8 shows the same conversion done by a graphing utility with a built in conversion routine If your calculator has a built in polar to rectangular conversion routine try it on V 2e and V2e i then reverse the process to see if you get back where you started For complex numbers in exponential polar form some calculators require 0 to be in radian mode for calculations Check your user s manual Write parts A C in rectangular form Compute the exact values for parts A and B for part C compute a and b for a bi to two decimal places A z M 2e C 9i B z 3e207i C z 6 49e 298i Let z V3 i and z 1 iV3 A Find z z and z z using the rectangular forms of z and Zp B Find z z and z z using the exponential polar forms of z and z 0 in degrees Assume the product and quotient exponent laws hold for e C Convert the results from part B back to
7. mber powers More importantly the theorem is the basis for the nth root theorem which enables us to find all n nth roots of any complex number real or imaginary
8. ola ellipse or a parabola A e 0 4 B e 1 C e 1 6 It is instructive to explore the graph for other positive val ues of e Conic Sections Using a graphing utility graph the equation SECTION 7 6 Complex Numbers in Rectangular and Polar Forms e Rectangular Form S e Polar Form e Multiplication and Division in Polar Form e Historical Note Utilizing polar concepts studied in the last section we now show how complex num bers can be written in polar form which can be very useful in many applications A brief review of Section 1 5 on complex numbers should prove helpful before pro ceeding further 556 e Rectangular Form Imaginary axis Real axis FIGURE 1 Complex plane EXAMPLE 1 Solution Matched Problem 1 EXPLORE DISCUSS 1 7 Additional Topics in Trigonometry Recall from Section 1 5 that a complex number is any number that can be written in the form a bi where a and b are real numbers and i is the imaginary unit Thus associated with each complex number a bi is a unique ordered pair of real numbers a b and vice versa For example 3 5i corresponds to 3 5 Associating these ordered pairs of real numbers with points in a rectangular coor dinate system we obtain a complex plane see Fig 1 When complex numbers are associated with points in a rectangular coordinate system we refer to the x axis as the real axis and the y axis as the imaginary axis The complex number
9. re as a polar form for a complex number In fact some graphing utilities display the polar form of x iy this way see Fig 3 where 0 is in radians Since cos 0 and sin 0 are both periodic with period 277 we have 1 41e rc 7912 cos 0 2km cos 0 k any integer sin 0 2kq sin 0 Thus we can write a more general polar form for a complex number z x iy as FIGURE 3 1 i 1 41e given below and observe that re is periodic with period 2km k any integer DEFINITION 1 General Polar Form of a Complex Number For k any integer N x iy r cos 0 2km i sin 0 2kr ygi 8 2km The number r is called the modulus or absolute value of z and is denoted by mod z or z The polar angle that the line joining z to the origin makes with the polar axis is called the argument of z and is denoted by arg z From Figure 2 we see the following relationships DEFINITION 2 Modulus and Argument for z x iy modz r Vx y Never negative arg z 0 2km k any integer where sin 0 y r and cos 0 x r The argument 6 is usually chosen so that 180 lt 6 S 180 or T 0 x m 558 7 Additional Topics in Trigonometry EXAMPLE 2 From Rectangular to Polar Form Write parts A C in polar form 0 in radians 7 lt 0 lt m Compute the modulus and arguments for parts A and B exactly compute the modulus and argument for part C to two decimal places A z 1 i B z V3 i C z 5 2i
10. rectangular form and compare with the results in part A You will now see a particular advantage of representing complex numbers in polar form Multiplication and division become very easy Theorem 1 provides the reason The exponential polar form of a complex number obeys the product and quotient rules for exponents b b b and b b b 7 6 Complex Numbers in Rectangular and Polar Forms 561 Theorem 1 Products and Quotients in Polar Form If z r e and z re then i ee CU i0 fie ie ea MORES Po 2 Ve f ee We establish the multiplication property and leave the quotient property for Prob lem 34 in Exercise 7 6 ZZ nevre r r cos 0 i sin 0 cos 0 i sin 0 r r cos 0 cos 0 i cos 0 sin 0 i sin 0 cos 0 sin 0 sin 0 r r cos 0 cos 0 sin 0 sin 05 i cos 0 sin 0 sin 0 cos 0 rr cos 0 0 i sin 0 0 r r e t EXAMPLE 4 Products and Quotients If z 8e and z 2e find A 212 B zj z Solution A z z 8e 263 r lt lt lt lt lt lt lt 7 Q 2g 05 30 16275 POT J 45 Zi Se B mA P E primi n in m i in 1 8 ji 459 30 i dels 2 pos Lorum ui MM J Matched Problem 4 1f 9e and z 3e find A 2125 B zj z Write in trigonometric form Multiply Use sum identities Write in exponential form 562 7 Additional Topic
11. s in Trigonometry e Historical Note EXERCISE 7 6 A In Problems 1 8 plot each set of complex numbers in a complex plane 1 A 3 4i B 2 i C 2i There is hardly an area in mathematics that does not have some imprint of the famous Swiss mathematician Leonhard Euler 1707 1783 who spent most of his productive life at the New St Petersburg Academy in Russia and the Prussian Academy in Berlin One of the most prolific writers in the history of the subject he is credited with mak ing the following familiar notations standard f x function notation e natural logarithmic base i imaginary unit V 1 For our immediate interest he is also responsible for the extraordinary relationship e cos 0 i sin 0 If we let 0 m we obtain an equation that relates five of the most important num bers in the history of mathematics e 1 0 Answers to Matched Problems 1 2 A V2 cos 3a 4 i sin 37 4 V2 B 2 cos m 3 i sin m 3 2e C 5 83 cos 2 11 i sin 2 11 5 83e771 A V2I B gt 3 A V2i B 5 7 4 A ziz 276 B z z 3e 1i i C 3 16 5 67i 6 A 20 9 B 4e C 2e 0 7 A 4e 59i B 3g i C 5e7 90 8 A 2e i B 3675 C Ge 2 A 4 i B 3 4 21 C 3i B 3 A 3 31 B 4 C 2 31 In Problems 9 12 change parts A C to polar form For 4 A 3 B 2 i1 C 4 4i Problems 9 and 10 choose 0 in degrees 180
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